A generalized Ramsey number in the minimum-colors sense is the least number 
of colors needed for an edge coloring of a graph 
in which every copy of a graph 
receives at least 
distinct colors. Copies need not be induced, and the coloring
need not be a proper coloring. The parameter satisfies
(Lane and Morrison 2026). This differs from the Ramsey
number convention, which fixes the number of colors and asks for the host size
forcing a monochromatic copy.
For each fixed 
,
Lane and Morrison (2026, Corollary 3) prove
 |
where 
is a complete graph, 
is a cycle graph, and 
denotes little-O notation as 
. Their Theorem 2 gives the same asymptotic
value when every cycle graph with length between fixed
integers 
and 
must receive at least three colors. For 
, the condition is precisely a proper
coloring of the graph edges of 
, so 
is its edge
chromatic number.
Lane and Morrison (2026) also prove upper bounds for even cycles in the complete bipartite graph 
using conflict-free hypergraph
matchings.
